Average Error: 23.8 → 11.6
Time: 39.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.9183315012152136 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right) \cdot \left(\left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]
\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 3.9183315012152136 \cdot 10^{+117}:\\
\;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right) \cdot \left(\left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right)\right)}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r4114162 = alpha;
        double r4114163 = beta;
        double r4114164 = r4114162 + r4114163;
        double r4114165 = r4114163 - r4114162;
        double r4114166 = r4114164 * r4114165;
        double r4114167 = 2.0;
        double r4114168 = i;
        double r4114169 = r4114167 * r4114168;
        double r4114170 = r4114164 + r4114169;
        double r4114171 = r4114166 / r4114170;
        double r4114172 = 2.0;
        double r4114173 = r4114170 + r4114172;
        double r4114174 = r4114171 / r4114173;
        double r4114175 = 1.0;
        double r4114176 = r4114174 + r4114175;
        double r4114177 = r4114176 / r4114172;
        return r4114177;
}
↓
double f(double alpha, double beta, double i) {
        double r4114178 = alpha;
        double r4114179 = 3.9183315012152136e+117;
        bool r4114180 = r4114178 <= r4114179;
        double r4114181 = beta;
        double r4114182 = r4114181 - r4114178;
        double r4114183 = r4114178 + r4114181;
        double r4114184 = 2.0;
        double r4114185 = i;
        double r4114186 = r4114184 * r4114185;
        double r4114187 = r4114183 + r4114186;
        double r4114188 = r4114182 / r4114187;
        double r4114189 = 2.0;
        double r4114190 = r4114189 + r4114187;
        double r4114191 = r4114188 / r4114190;
        double r4114192 = r4114191 * r4114183;
        double r4114193 = 1.0;
        double r4114194 = r4114192 + r4114193;
        double r4114195 = r4114194 * r4114194;
        double r4114196 = r4114194 * r4114195;
        double r4114197 = cbrt(r4114196);
        double r4114198 = r4114197 / r4114189;
        double r4114199 = r4114189 / r4114178;
        double r4114200 = 8.0;
        double r4114201 = r4114178 * r4114178;
        double r4114202 = r4114201 * r4114178;
        double r4114203 = r4114200 / r4114202;
        double r4114204 = 4.0;
        double r4114205 = r4114204 / r4114201;
        double r4114206 = r4114203 - r4114205;
        double r4114207 = r4114199 + r4114206;
        double r4114208 = r4114207 / r4114189;
        double r4114209 = r4114180 ? r4114198 : r4114208;
        return r4114209;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 3.9183315012152136e+117

    1. Initial program 14.4

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    2. Using strategy rm

    3. Applied *-un-lft-identity14.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \color{blue}{1 \cdot 2.0}} + 1.0}{2.0}\]

    4. Applied *-un-lft-identity14.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + 1 \cdot 2.0} + 1.0}{2.0}\]

    5. Applied distribute-lft-out14.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

    6. Applied *-un-lft-identity14.4

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    7. Applied times-frac4.1

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

    8. Applied times-frac4.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

    9. Simplified4.1

      \[\leadsto \frac{\color{blue}{\left(\beta + \alpha\right)} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
    10. Using strategy rm

    11. Applied div-inv4.1

      \[\leadsto \frac{\left(\beta + \alpha\right) \cdot \color{blue}{\left(\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)} + 1.0}{2.0}\]

    12. Applied associate-*r*4.1

      \[\leadsto \frac{\color{blue}{\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]
    13. Using strategy rm

    14. Applied add-cbrt-cube4.1

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)\right) \cdot \left(\left(\left(\beta + \alpha\right) \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}\right) \cdot \frac{1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0\right)}}}{2.0}\]

    15. Simplified4.1

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(\left(1.0 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \left(1.0 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)\right) \cdot \left(1.0 + \left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right)}}}{2.0}\]

    if 3.9183315012152136e+117 < alpha

    1. Initial program 59.5

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

    2. Taylor expanded around -inf 40.1

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

    3. Simplified40.1

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.9183315012152136 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt[3]{\left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right) \cdot \left(\left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right) \cdot \left(\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{2.0 + \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \left(\alpha + \beta\right) + 1.0\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019134 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))